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Some New Integral Inequalities for Functions Whose Derivatives of Absolute Values Are s-Convex

M. Emin Özdemir*, Alper Ekinci

Uludag University, Education Faculty, Bursa, Turkey

Abstract: In this paper, we prove some new inequalities for the functions whose derivatives absolute values are s-convex by dividing the interval to equal even sub-intervals. We obtain some new results involving intermediate values of in by using some classical inequalities like Hermite-Hadamard, Hölder and Power-Mean.

Keywords: s-convex functions, Hermite-Hadamard Inequality, power-mean inequality, hölder inequality

Received March 04, 2019; Revised April 08, 2019; Accepted May 26, 2019

Citation: M. Emin Özdemir, Alper Ekinci, Some New Integral Inequalities for Functions Whose Derivatives of Absolute Values Are s-Convex. Turkish Journal of Analysis and Number Theory, 7 (3). 70-76. January 2019. doi:10.12691/tjant-7-3-3

1. Introduction

The function is said to be convex, if we have

for all and Geometrically, this means that if and are three distinct points on the graph of with between and , then is on or below chord A huge amount of the researchers interested in this definition and there are several papers based on convexity. See the papers [1,2,3,4,5,6,7,8,9,10].

Let be a convex function and let with The following double inequality;

is known in the literature as Hadamard’s inequality [5]. Both inequalities hold in the reversed direction if is concave.

In [2] Hudzik and Maligranda introduced following definition:

Definition 1.1. Let be fixed real number. A function is said to be convex (in the second sense), or that belongs to the class , if

holds for all with .

Orlicz gave the following definition of convexity in the first sense in [3]:

Definition 1.2. Let be fixed real number. A function is said to be convex (in the first sense), or that belongs to the class , if

holds for all with .

It is clear that convexity mean just the convexity when . In [11], Dragomir and Fitzpatrick proved the following variant of Hadamard’s inequality which hold for convex functions in the second sense:

Theorem 1.3. Suppose that is an convex function in the second sense, where and let , . If , then the following inequalities hold:

[picture] | (1.1)

The constant is the best possible in the second inequality in (1.1).

In a recent paper [1], Latif and Dragomir proved following Theorems:

Theorem 1.4. Let be a differentiable function on such that where with If is convex on then the following inequality holds:

[picture] | (1.2)

Theorem 1.5. Let be a differentiable function on such that where with If is convex on for some fixed then the following inequality holds:

[picture] | (1.3)

where

Theorem 1.6. Let be a differentiable function on such that where with If is convex on for some fixed then the following inequality holds:

[picture] | (1.4)

Theorem 1.7. Let be a differentiable function on such that where with If is concave on for some fixed then the following inequality holds:

[picture] | (1.5)

where

Theorem 1.8. Let be a differentiable function on such that where with If is concave on for some fixed then the following inequality holds:

[picture] | (1.6)

The main aim of this paper is to establish some new inequalities involving values of at intermediate points of interval for functions whose absolute values of derivatives are s-convex and s-concave.

2. Main Results

We need following lemma to prove our main Theorems:

Lemma 2.1. [4]Let be a differentiable function on where with If and is an odd number then the following equality holds:

[picture] | (2.1)

Theorem 2.2. Let be a differentiable function on where with If is convex on in the second sense and is an odd number then the following inequality holds:

[picture] | (2.2)

Proof. By using Lemma 2.1 and properties of modulus, we have

By using the convexity of we obtain

Which completes the proof.

Corollary 2.3. If we choose in (2.2) we obtain the following result:

Remark 2.4. If we choose and in (2.2) , this inequality reduces to (1.2).

Theorem 2.5. Let be a differentiable function on where with If is convex on in the second sense for some fixed and is an odd number, then the following inequality holds:

[picture] | (2.3)

where

Proof. From Lemma 2.1 and by using the Hölder inequality, we have

[picture] | (2.4)

Since is convex on , we have

Similarly,

By using the last two inequalities in (), we obtain the desired result.

Corollary 2.6. If we choose in (2.3), we obtain the following result:

Remark 2.7. If we choose and in (2.3), this inequality reduces to (1.3).

Theorem 2.8. Let be a differentiable function on such that where with If is convex on in the second sense for some fixed and is an odd number, then the following inequality holds:

[picture] | (2.5)

Proof. From Lemma 2.1 and by using Power mean inequality, we have

[picture] | (2.6)

Since is convex on in the second sense, we have

Similarly,

Using the last two inequalities in (2.6), we get the result.

Corollary 2.9. If we choose in (2.5), we obtain the following result:

Remark 2.10. If we choose and in (2.3), this inequality reduces to (1.4).

Theorem 2.11. Let be a differentiable function on such that where with If is concave on in the second sense for some fixed and is an odd number, then the following inequality holds:

[picture] | (2.7)

where

Proof. From Lemma 2.1 and using the Hölder inequality for and we have

[picture] | (2.8)

Since is s-concave on and by using the Hadamard inequality for concave functions, we have

and similarly,

Using these two inequalities in we get the desired result.

Corollary 2.12. If we choose in we obtain the following result:

Corollary 2.13. Under the conditions of Theorem 2.11 and assume that is a linear function, the following inequality holds:

Proof. It follows directly from Theorem 2.11 and linearity of

Remark 2.14. If we choose and in this inequality reduces to

References

[1]   | M.A. Latif and S. S. Dragomir, New Inequalities of Hermite-Hadamard Type For Functions Whose Derivatives In Absolute Value are Convex With Applications to Special Means and to General Quadrature Formula, J. Inequal. Pure and Appl. Math., 9, (4), (2007), Article 96.
[2]   | H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math, 48 (1994), 100-111.
[3]   | W. Orlicz, A note on modular spaces 1, Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phys., 9 (1961), 157-162.
[4]   | M. E. Özdemir, A. Ekinci and A. O. Akdemir, Some New Integral Inequalities for Functions Whose Derivatives of Absolute Values are convex and concave, TWMS Journal of Pure and Applied Mathematics (2019) Accepted.
[5]   | J. Hadamard, Etude sur les propriétés des fonctions entiéres et en particulier dune fonction considerée par Riemann, J. Math Pures Appl., 58 (1893), 171-215.
[6]   | K.L. Tseng, S.R. Hwang and S. S. Dragomir, Fejer-type inequalities (I). Journal of Inequalities and Applications Volume 2010, Article ID 531976.
[7]   | S. S. Dragomir and C. E. M. Pearce, Selected Topic on Hermite- Hadamard Inequalities and Applications, Melbourne and Adelaide, December, 2000.
[8]   | U. S. Kirmac, K. Bakula, M. E. Özdemir and J. Pecaric, Hadamard-type inequalities for s-convex functions, Appl. Math. Comput., 193(1) (2007) 26-35.
[9]   | H. Kavurmaci, M. Avci and M. E. Özdemir, New inequalities of Hermite-Hadamard type for convex functions with applications, Journal of Inequalities and Applications (2011).
[10]   | B.G. Pachpatte, On some inequalities for convex functions, RGMIA Research Report Collection, 6(E) (2003).
[11]   | S.S. Dragomir, S. Fitzpatrick, The Hadamard’s inequality for s-convex functions in the second sense, Demonstratio Math., 32 (4) (1999), 687–696.